Averages: Simple & Weighted Average in Business

1. Simple Average (Arithmetic Mean)

The simple average is obtained by dividing the sum of all numerical observations by the total number of observations:

Mean (x̄) = (Sum of all observations) ÷ (Total number of observations)
x̄ = (X₁ + X₂ + … + X_N) ÷ N = (ΣX) ÷ N

Properties of Arithmetic Mean:

1. The algebraic sum of deviations of all observations from their arithmetic mean is always zero: Σ(X - x̄) = 0.
2. The sum of squared deviations from the mean is minimum: Σ(X - x̄)² = minimum.

2. Weighted Average: Commercial Superiority

In real commercial contexts, quantities do not have equal economic significance. For example, buying 10 units at ₹50 and 1,000 units at ₹70 cannot be averaged by simply doing (50 + 70) ÷ 2 = 60. We must account for the lot sizes:

Weighted Mean (x̄_w) = Σ(W × X) ÷ ΣW
x̄_w = (W₁X₁ + W₂X₂ + … + W_nX_n) ÷ (W₁ + W₂ + … + W_n)

3. Practical Inventory Costing Problem

Question: A textile retailer makes three bulk purchases of fabric during a quarter:

• Lot 1: 300 meters at ₹45 per meter
• Lot 2: 500 meters at ₹52 per meter
• Lot 3: 200 meters at ₹60 per meter

Determine the weighted average cost per meter.

Step-by-Step Solution:

1. Calculate total cost for each purchase lot:

   • Lot 1: 300 meters × ₹45 = ₹13,500
   • Lot 2: 500 meters × ₹52 = ₹26,000
   • Lot 3: 200 meters × ₹60 = ₹12,000

2. Sum the total quantities and total expenditures:

   • Total Meters (ΣW) = 300 + 500 + 200 = 1,000 meters
   • Total Cost (ΣWX) = ₹13,500 + ₹26,000 + ₹12,000 = ₹51,500

3. Compute the Weighted Average Cost per meter:

   • Weighted Average = Total Cost ÷ Total Meters
   • Weighted Average = ₹51,500 ÷ 1,000 meters
   • Weighted Average Cost = ₹51.50 per meter

(Note: Simple unweighted average would have been (45 + 52 + 60) ÷ 3 = ₹52.33, which incorrectly overstates inventory cost by ₹0.83 per meter).